Question

What is the value of the line integral of a magnetic field around a closed loop according to Ampère’s Circuital Law?

• A. \(0\)
• B. \(\mu_0 I\)
• C. \(\dfrac{I}{\mu_0}\)
• D. \(B \times I\)

Hint:

{Ampère’s Law:} \(\displaystyle \oint \vec{B}\cdot d\vec{l} = \mu_0 I\). The circulation of magnetic field around a closed loop depends on the enclosed current.

Answer 1

The Correct Option is B

Concept: Ampère’s Circuital Law states that the line integral of the magnetic field \(\vec{B}\) around any closed path is equal to the permeability of free space multiplied by the total current enclosed by that path. The mathematical form of Ampère’s Law is: \[ \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}} \] where

• \(\vec{B}\) = Magnetic field
• \(d\vec{l}\) = Infinitesimal length element of the closed loop
• \(\mu_0\) = Permeability of free space
• \(I_{\text{enc}}\) = Current enclosed by the loop

Step 1: Identify the result of the line integral.} The line integral of the magnetic field around a closed path equals \(\mu_0\) multiplied by the enclosed current.
Step 2: Write the final value.} \[ \oint \vec{B}\cdot d\vec{l} = \mu_0 I \] Thus, the value of the line integral is \(\mu_0 I\).